Calculus Examples

${(2 x + 1)}^{- 1} (3 x - 1)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(2 x + 1)}^{- 1}$ and $g (x) = 3 x - 1$ .

${(2 x + 1)}^{- 1} \frac{d}{d x} [3 x - 1] + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

${(2 x + 1)}^{- 1} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

${(2 x + 1)}^{- 1} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

${(2 x + 1)}^{- 1} (3 \cdot 1 + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 2.4

Multiply $3$ by $1$ .

${(2 x + 1)}^{- 1} (3 + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

${(2 x + 1)}^{- 1} (3 + 0) + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

${(2 x + 1)}^{- 1} \cdot 3 + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 2.6.2

Move $3$ to the left of ${(2 x + 1)}^{- 1}$ .

$3 \cdot {(2 x + 1)}^{- 1} + (3 x - 1) \frac{d}{d x} [{(2 x + 1)}^{- 1}]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 2 x + 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $2 x + 1$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [2 x + 1])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 1$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- u^{- 2} \frac{d}{d x} [2 x + 1])$

Step 3.3

Replace all occurrences of $u$ with $2 x + 1$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- {(2 x + 1)}^{- 2} \frac{d}{d x} [2 x + 1])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- {(2 x + 1)}^{- 2} (\frac{d}{d x} [2 x] + \frac{d}{d x} [1]))$

Step 4.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- {(2 x + 1)}^{- 2} (2 \frac{d}{d x} [x] + \frac{d}{d x} [1]))$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- {(2 x + 1)}^{- 2} (2 \cdot 1 + \frac{d}{d x} [1]))$

Step 4.4

Multiply $2$ by $1$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- {(2 x + 1)}^{- 2} (2 + \frac{d}{d x} [1]))$

Step 4.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- {(2 x + 1)}^{- 2} (2 + 0))$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $2$ and $0$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- {(2 x + 1)}^{- 2} \cdot 2)$

Step 4.6.2

Multiply $2$ by $- 1$ .

$3 {(2 x + 1)}^{- 1} + (3 x - 1) (- 2 {(2 x + 1)}^{- 2})$

Step 5

Simplify.

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Step 5.1

Reorder terms.

$- 2 {(2 x + 1)}^{- 2} (3 x - 1) + 3 {(2 x + 1)}^{- 1}$

Step 5.2

Simplify each term.

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Step 5.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 \frac{1}{{(2 x + 1)}^{2}} (3 x - 1) + 3 {(2 x + 1)}^{- 1}$

Step 5.2.2

Combine $- 2$ and $\frac{1}{{(2 x + 1)}^{2}}$ .

$\frac{- 2}{{(2 x + 1)}^{2}} (3 x - 1) + 3 {(2 x + 1)}^{- 1}$

Step 5.2.3

Move the negative in front of the fraction.

$- \frac{2}{{(2 x + 1)}^{2}} (3 x - 1) + 3 {(2 x + 1)}^{- 1}$

Step 5.2.4

Apply the distributive property.

$- \frac{2}{{(2 x + 1)}^{2}} (3 x) - \frac{2}{{(2 x + 1)}^{2}} \cdot - 1 + 3 {(2 x + 1)}^{- 1}$

Step 5.2.5

Multiply $- \frac{2}{{(2 x + 1)}^{2}} (3 x)$ .

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Step 5.2.5.1

Multiply $3$ by $- 1$ .

$- 3 \frac{2}{{(2 x + 1)}^{2}} x - \frac{2}{{(2 x + 1)}^{2}} \cdot - 1 + 3 {(2 x + 1)}^{- 1}$

Step 5.2.5.2

Combine $- 3$ and $\frac{2}{{(2 x + 1)}^{2}}$ .

$\frac{- 3 \cdot 2}{{(2 x + 1)}^{2}} x - \frac{2}{{(2 x + 1)}^{2}} \cdot - 1 + 3 {(2 x + 1)}^{- 1}$

Step 5.2.5.3

Multiply $- 3$ by $2$ .

$\frac{- 6}{{(2 x + 1)}^{2}} x - \frac{2}{{(2 x + 1)}^{2}} \cdot - 1 + 3 {(2 x + 1)}^{- 1}$

Step 5.2.5.4

Combine $\frac{- 6}{{(2 x + 1)}^{2}}$ and $x$ .

$\frac{- 6 x}{{(2 x + 1)}^{2}} - \frac{2}{{(2 x + 1)}^{2}} \cdot - 1 + 3 {(2 x + 1)}^{- 1}$

Step 5.2.6

Multiply $- \frac{2}{{(2 x + 1)}^{2}} \cdot - 1$ .

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Step 5.2.6.1

Multiply $- 1$ by $- 1$ .

$\frac{- 6 x}{{(2 x + 1)}^{2}} + 1 \frac{2}{{(2 x + 1)}^{2}} + 3 {(2 x + 1)}^{- 1}$

Step 5.2.6.2

Multiply $\frac{2}{{(2 x + 1)}^{2}}$ by $1$ .

$\frac{- 6 x}{{(2 x + 1)}^{2}} + \frac{2}{{(2 x + 1)}^{2}} + 3 {(2 x + 1)}^{- 1}$

Step 5.2.7

Move the negative in front of the fraction.

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{2}{{(2 x + 1)}^{2}} + 3 {(2 x + 1)}^{- 1}$

Step 5.2.8

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{2}{{(2 x + 1)}^{2}} + 3 \frac{1}{2 x + 1}$

Step 5.2.9

Combine $3$ and $\frac{1}{2 x + 1}$ .

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{2}{{(2 x + 1)}^{2}} + \frac{3}{2 x + 1}$

Step 5.3

To write $\frac{3}{2 x + 1}$ as a fraction with a common denominator, multiply by $\frac{2 x + 1}{2 x + 1}$ .

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{3}{2 x + 1} \cdot \frac{2 x + 1}{2 x + 1} + \frac{2}{{(2 x + 1)}^{2}}$

Step 5.4

Write each expression with a common denominator of ${(2 x + 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{3}{2 x + 1}$ by $\frac{2 x + 1}{2 x + 1}$ .

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{3 (2 x + 1)}{(2 x + 1) (2 x + 1)} + \frac{2}{{(2 x + 1)}^{2}}$

Step 5.4.2

Raise $2 x + 1$ to the power of $1$ .

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{3 (2 x + 1)}{{(2 x + 1)}^{1} (2 x + 1)} + \frac{2}{{(2 x + 1)}^{2}}$

Step 5.4.3

Raise $2 x + 1$ to the power of $1$ .

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{3 (2 x + 1)}{{(2 x + 1)}^{1} {(2 x + 1)}^{1}} + \frac{2}{{(2 x + 1)}^{2}}$

Step 5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{3 (2 x + 1)}{{(2 x + 1)}^{1 + 1}} + \frac{2}{{(2 x + 1)}^{2}}$

Step 5.4.5

Add $1$ and $1$ .

$- \frac{6 x}{{(2 x + 1)}^{2}} + \frac{3 (2 x + 1)}{{(2 x + 1)}^{2}} + \frac{2}{{(2 x + 1)}^{2}}$

Step 5.5

Combine the numerators over the common denominator.

$\frac{- 6 x + 3 (2 x + 1)}{{(2 x + 1)}^{2}} + \frac{2}{{(2 x + 1)}^{2}}$

Step 5.6

Combine the numerators over the common denominator.

$\frac{- 6 x + 3 (2 x + 1) + 2}{{(2 x + 1)}^{2}}$

Step 5.7

Simplify each term.

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Step 5.7.1

Apply the distributive property.

$\frac{- 6 x + 3 (2 x) + 3 \cdot 1 + 2}{{(2 x + 1)}^{2}}$

Step 5.7.2

Multiply $2$ by $3$ .

$\frac{- 6 x + 6 x + 3 \cdot 1 + 2}{{(2 x + 1)}^{2}}$

Step 5.7.3

Multiply $3$ by $1$ .

$\frac{- 6 x + 6 x + 3 + 2}{{(2 x + 1)}^{2}}$

Step 5.8

Add $- 6 x$ and $6 x$ .

$\frac{0 + 3 + 2}{{(2 x + 1)}^{2}}$

Step 5.9

Add $0$ and $3$ .

$\frac{3 + 2}{{(2 x + 1)}^{2}}$

Step 5.10

Add $3$ and $2$ .

$\frac{5}{{(2 x + 1)}^{2}}$